Riemannian Generative Decoder
Pith reviewed 2026-05-19 07:19 UTC · model grok-4.3
The pith
A decoder network trained jointly with Riemannian optimization learns latents on any Riemannian manifold.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The Riemannian generative decoder learns manifold-valued latents by jointly training a decoder network while optimizing the latent points using Riemannian optimization, thereby simplifying the manifold constraint and avoiding density estimation.
What carries the argument
The Riemannian generative decoder: a decoder network that maps from latents to data while the latents are optimized directly via Riemannian methods to enforce the chosen manifold geometry.
Load-bearing premise
A decoder network alone suffices to recover meaningful manifold-valued representations when trained jointly with Riemannian optimization on the latent points, without needing density estimation from an encoder.
What would settle it
If optimized latent points fail to remain on the prescribed manifold or produce reconstructions that ignore the manifold geometry when applied to a new manifold, the central claim would not hold.
read the original abstract
Euclidean representations distort data with intrinsic non-Euclidean structure. While Riemannian representation learning offers a solution by embedding data onto matching manifolds, it typically relies on an encoder to estimate densities on chosen manifolds. This involves optimizing numerically brittle objectives, potentially harming model training and quality. To completely circumvent this issue, we introduce the Riemannian generative decoder, a unifying approach for finding manifold-valued latents on any Riemannian manifold. Latents are learned with a Riemannian optimizer while jointly training a decoder network. By discarding the encoder, we vastly simplify the manifold constraint compared to current approaches which often only handle few specific manifolds. We validate our approach on three case studies -- a synthetic branching diffusion process, human migrations inferred from mitochondrial DNA, and cells undergoing a cell division cycle -- each showing that learned representations respect the prescribed geometry and capture intrinsic non-Euclidean structure. Our method requires only a decoder, is compatible with existing architectures, and yields interpretable latent spaces aligned with data geometry. Code available on https://github.com/yhsure/riemannian-generative-decoder.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces the Riemannian generative decoder, a method for learning manifold-valued latent representations on arbitrary Riemannian manifolds. It does so by jointly training a decoder network while directly optimizing per-point latent variables with a Riemannian optimizer, thereby discarding the encoder and its associated density estimation. The approach is validated through three qualitative case studies on a synthetic branching diffusion process, human migrations inferred from mitochondrial DNA, and cells in a division cycle, with the claim that the resulting latents respect the prescribed geometry and capture intrinsic non-Euclidean structure.
Significance. If the joint optimization reliably produces coherent, geometry-respecting latents, the method would simplify Riemannian representation learning and extend it beyond the limited manifolds handled by current encoder-based techniques. The public code release and compatibility with existing decoder architectures are clear strengths that aid reproducibility. However, the current validation relies solely on qualitative visualizations without quantitative metrics or baselines, which limits assessment of whether the claimed training stability and meaningful representations are achieved.
major comments (2)
- [Method] The central claim that discarding the encoder 'vastly simplify[s] the manifold constraint' and suffices for arbitrary manifolds rests on the assumption that reconstruction-driven Riemannian updates on latents alone recover global structure without density estimation or a learned prior. This is not supported by additional analysis or theoretical justification in the method description, leaving open the possibility that latents minimize reconstruction loss while failing to reflect intrinsic geometry (e.g., uniform coverage on branching or cyclic data).
- [Experiments] The three case studies (synthetic branching diffusion, mitochondrial DNA migrations, cell-cycle data) are presented only qualitatively. No quantitative metrics, error bars, ablation studies, or comparisons to encoder-based Riemannian baselines are reported, which undermines the ability to verify the claimed improvements in training stability and representation quality.
minor comments (2)
- [Abstract] The abstract and introduction could more explicitly define the Riemannian optimizer used for the latent variables and how it interfaces with the decoder gradients.
- [Figures] Figure captions for the case-study visualizations would benefit from additional detail on how the manifold geometry is visualized and what specific aspects demonstrate respect for the prescribed structure.
Simulated Author's Rebuttal
We thank the referee for their constructive comments on our manuscript. We address the major comments point by point below.
read point-by-point responses
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Referee: [Method] The central claim that discarding the encoder 'vastly simplify[s] the manifold constraint' and suffices for arbitrary manifolds rests on the assumption that reconstruction-driven Riemannian updates on latents alone recover global structure without density estimation or a learned prior. This is not supported by additional analysis or theoretical justification in the method description, leaving open the possibility that latents minimize reconstruction loss while failing to reflect intrinsic geometry (e.g., uniform coverage on branching or cyclic data).
Authors: The approach is motivated by the observation that directly optimizing manifold-valued latents via Riemannian methods while training a decoder for reconstruction allows the model to operate on arbitrary manifolds without requiring a parametric density estimator on the manifold. The decoder must accurately map from the prescribed manifold to the data, which in practice encourages the latents to align with the data's intrinsic geometry, as seen in the case studies. We acknowledge that the manuscript would benefit from a more explicit discussion of this assumption and its limitations. We will revise the method section to include additional explanation of why reconstruction-driven updates suffice in this setting, along with a brief analysis of potential failure modes. revision: yes
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Referee: [Experiments] The three case studies (synthetic branching diffusion, mitochondrial DNA migrations, cell-cycle data) are presented only qualitatively. No quantitative metrics, error bars, ablation studies, or comparisons to encoder-based Riemannian baselines are reported, which undermines the ability to verify the claimed improvements in training stability and representation quality.
Authors: We agree that quantitative evaluation would strengthen the claims. The current case studies focus on qualitative demonstration across diverse manifolds where suitable quantitative baselines for arbitrary Riemannian manifolds are not always straightforward to define. In the revision we will add quantitative metrics, including reconstruction error on held-out points, comparisons against Euclidean decoder baselines, and simple measures of manifold adherence such as average geodesic deviation. Results will be reported with error bars from multiple random seeds, and we will include a limited ablation on the choice of Riemannian optimizer. revision: yes
Circularity Check
No significant circularity in the proposed decoder-only Riemannian optimization
full rationale
The paper presents a methodological proposal for learning manifold-valued latents by jointly optimizing them with a Riemannian optimizer and a decoder network while discarding the encoder. This is framed as a practical simplification of manifold constraints rather than a derivation of new results from previously fitted quantities or self-referential definitions. No equations, self-citations, or ansatzes are shown in the provided text that reduce any central claim to its own inputs by construction. The approach relies on standard joint optimization and empirical validation on case studies, remaining self-contained without load-bearing circular steps.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Riemannian manifolds admit well-defined exponential and logarithmic maps that can be used inside an optimizer.
discussion (0)
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